Math 307 Spring 2003 Hentzel Time 1 10

Math 307 Spring, 2003 Hentzel Time: 1: 10 -2: 00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515 -294 -8141 E-mail: hentzel@iastate. edu http: //www. math. iastate. edu/hentzel/class. 307. ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher

Monday, Apr 21 Chapter 7. 3 Page 324 Problems 16, 20. 34, 36 Main Idea: You want lots of eigen vectors. You might not get all you want. Key Words: Algebraic Multiplicity, Geometric Multiplicity, Eigen Space Goal: Learn to expect additional eigen vectors for multiple roots, but accept graciously the possibility that they do not exist.

Previous Assignment Friday, April 18 Chapter 7. 2 Page 310 Problems 6, 8, 10, 20

Page 310 Problem 6 Find all real eigenvalues, with their algebraic multiplicities. A= |1 2 | |3 4 | Det [A-x I ] = Det | 1 -x 2 | = (1 -x)(4 -x) – 6 | 3 4 -x | 2 = x -5 x – 2 5 +/- Sqrt [33] x = --------2 x = -0. 372281 x = 5. 37228

Page 310 Problem 8 Find all real eigenvalues, with their algebraic multiplicities. | -1 A = | -1 -1 -1 |

| -2 -x -1 -1 | Det [A-x. I] = x | 0 -1 0 | | -2 -1 -1 -x | | -2 -x Det [A-x. I] = -x | | -2 -1 | | -1 -x | Det [A-x. I] = -x [ (-2 -x)(-1 -x) - 2 ] 2 De t[A-x. I] = -x [ 2+2 x+x+x - 2 ] 2 Det [A-x. I] = -x [ x + 3 x ] Det [A-x. I] = -x 2[ x+3] The eigen values are 0, 0, -3.

Page 310 Problem 10 Find all real eigenvalues, with their algebraic multiplicities. | -3 0 4 | A = | 0 -1 0 | | -2 7 3 |

2 Det [A - x I ] = (-1 -x)( -9 + 3 x - 3 x + 8 ) Det [A - x I ] = (-1 -x)( x 2 -1) = -(x+1)2 (x-1) Eigen values are -1, 1

Page 310 Problem 20 Consider a 2 x 2 matrix A with two distinct real eigenvalues c 1 and c 2. Express Det [A] in terms of c 1 and c 2. Do the same for the trace of A. A= | a b | | c d | Det | a-x b | = (a-x)(d-x) - bc | c d-x |

2 Det[A-x. I] = x -(a+d)x + (ad-bc) 2 = x - trace[A] x + Det[A]. 2 = (x-c 1)(x-c 2) = x - (c 1+c 2) x + c 1 c 2 So c 1+c 2 = trace[A] and c 1 c 2 = Det[A].

New Material: The Characteristic polynomial of a matrix A is Det[A-x. I].

Theorem. The Characteristic polynomial is invariant under similarity.

The Characteristic Polynomial of P -1 A P = Det [ P -1 A P - x I ] = -1 Det [ P (A-x I) P ] = Det [ P -1 Det[A-x I] Det[P] = -1 Det [ P P] Det[A-x. I] = Det [A-x. I] = The Characteristic Polynomial of A.

Corollary: Similar matrices have the same eigen values. If c, V; are an eigen value and eigen vector of A, then c, P -1 V; are an eigen value and vector of P -1 A P.

Proof: P -1 A P P -1 V = P -1 A V = P -1 c V = c P -1 V.

Definition: The Algebraic multiplicity is the number of times c appears as a root of the characteristic polynomial. The Geometric multiplicity of the eigen value c is the number of linearly independent eigen vectors with eigen value c.

Find the eigen values and eigen vectors of | 0 1 0 | A=| 0 0 1 | | 0 0 0 |

| 0 -x 1 0 | 3 Det [A - x I ] = | 0 0 -x 1 | = -x | 0 0 0 -x | The eigen values are 0, 0, 0.

To find the corresponding eigen vectors. | 0 1 0| |1| | 0 0 1 | has null space | 0 | | 0 0 0| |0| So 0 has algebraic multiplicity 3 and geometric multiplicity 1 for A.

Page 321 Example 7. Solve the recursive dependence relation X(t+1) = A x(t) | 750 | | 0 19 12 | Where X(0) = | 200 | and A =(1/20) | 16 0 0 | | 200 | | 0 10 0 |

The Eigen Values and Vectors are: | 9| | 2| |5| -3/5 |-12 | -2/5 | -4 | 1 |4| | 10 | | 5| |2| | 9 P = | -12 | 10 2 -4 5 5| 4 |. 2|

| -3/5 0 P -1. A. P = | 0 -2/5 | 0 0 | 750 | Solve P X = | 200 | 0| 0| 1| | 50 | X = | -100 | | 100 |

Xo = | 9| |2| |5| 50 |-12 | - 100 |-4 | + 100 | 4 | | 10 | |5| |2|

| 9| |2| |5| Xn = 50 (-3/5)n |-12 | - 100 (-2/5) n |-4 | +100| 4 | | 10 | |5| |2| The limiting situation is | 500 | | 400 | | 200 |

Theorem: Eigen Vectors for distinct Eigen Values are linearly independent.

Proof: Suppose that ci, Vi are eigen values and eigen vectors for distinct eigen values. Suppose that a 1 V 1 + a 2 V 2 + … + an Vn = 0 then a 1 c 1 n V 1 + a 2 c 2 n V 2+ … + a n cn n V n = 0 for all n. This can only happen when all ai are zero.